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The example I gave in comments has $E/G_1$ and $E/G_2$ compact (but not manifolds), at least if you use compact "blobs". More explicitly, let $E$ be the subset of $\mathbb{R}^2$ given by the union of …

I think Milnor answers your question in the paper where he proves the theorem you refer to ("On spaces having the homotopy type of a CW-complex", Trans. AMS, 90 (1959), 272-280): Whitehead had observ …

Can't you just take $X=[0,1]$ and $K=\{0,1\}$, with $$f(x,t)=\begin{cases}\frac{x}{1-t}&\mbox{if $t\leq 1-x$}\\ 1&\mbox{if $t\geq 1-x$},\end{cases}$$ so that $f(x,t)\to 1$ as $t\to 1$ unless $x=0$, in …

Expanding on my comment, if there are measurable cardinals then it follows from the results of A. Przeździecki, Measurable cardinals and fundamental groups of compact spaces. Fund. Math. 192 (2006), …

If $X^{\bullet}$ is an object of $\mathrm{Ch}^{\geq0}(\mathcal{A})$ and for each $i\geq0$ $X^i\to I^i$ is an embedding into an injective, then $X^{\bullet}$ embeds in $$I^{\bullet}:=\dots\to 0\to I^0\ …

I'm not sure I understand your intuition, but the statement is true. It's a special case of Shapiro's Lemma: if $H$ is a subgroup of $G$, $A$ is a $\mathbb{Z}H$-module, and $A^G$ is the coinduced modu …

$R[M]$ is just a permutation module, so, by Shapiro's Lemma, $H_{\ast}(G;R[M])$ is the direct sum, over a set of representatives $m$ of orbits of $G$ acting on $M$, of the homology $H_{\ast}(G_m;R)$ w …

How about adding another object $W$ to your example with an automorphism $\delta:W\to W$ with $\delta^{-1}=\delta$ and arrows $\theta,\phi:X\to W$ with $\delta\circ\theta=\phi=\theta\circ\gamma$? The …

If $G$ acts freely on a CW-complex, permuting the cells, then the stabilizer of a cell must be finite (and therefore trivial, as pointed out in the question). This can be shown by induction on the di …

If a finite $p$-group $P$ acts on a finite-dimensional $CW$-complex $X$ which is acyclic mod $p$, then the fixed point set $X^P$ is also acyclic mod $p$. This is a special case of "Smith theory" (see …

No. The projective model structure on chain complexes of modules over a ring is an abelian model category, and the homotopy category is the derived category, which is never abelian unless the ring is …

I'm afraid this is not very close to the case that you say you're most interested in ... maybe you want $F$ to preserve products? But let $A=k[x]/(x^2)$, and let $\mathscr{A}$ be the category $\operat …

As noted in the question, "Question 3" reduces to asking whether, if $H_i(G,M)=0$ for all $i>0$ and all $\mathbb{Z}G$-modules $M$, then $G$ must be trivial. But this is true, since if $G$ is non-triv …

I just stumbled across the answer to this in Fuchs' 2015 book on Abelian Groups. The papers Hill, Paul, Two problems of Fuchs concerning tor and hom, J. Algebra 19, 379-383 (1971). ZBL0228.20027. and …